How to get rid of YouTube desktop and go back to YouTube Moible
How to get rid YouTube desktop or how to get YouTube working when it's on the desktop version or get it back to Moible for iPad2
iPad 2, iOS 6.1.3
There is a Mobile link at the bottom of the web page, although I suspect you already found it is broken and doesn't work. To switch back to the mobile version you will have to clear the cookies and cache for Safari. You will have to log into any web sites you set to remember you again. Go to Settings/Safari/Clear Cookies and Data.
You will have to do the same in any other web browser you are using although the clear data setting may be in an other location. There is also a YouTube app available which is tuned for the iPad and does work a little better.